George Carlin quipped, "In baseball the object is to go home! And be safe!" Exploration of baseball history shows this quip provides the answer to the question.

By "home run" I think you mean the situation described Rule 5.05(a)(5), which states that a batter becomes a runner in the event of hitting "[a] fair ball over the fence or into the stands at a distance from home base or 250 feet or more" and is automatically "entitle[d] to a home run when he shall have touched all bases legally." (A hypothetical shorter hit would just be a double. The closest fence in MLB today is 302 feet, the right foul line at Fenway Park, and new stadiums are supposed to have a minimum distance of 325 feet, so in fact you can't do this.) In the actual fan experience, this is quite exciting, everyone stands and hopes for the ball to leave the park or stay in (depending on their allegiances) and the announcer will proclaim, "He hit it over the fence! It's a home run!"

Note this is not the definition of a "home run" in baseball. Rule 9.06(a) specifies that a hit "is a home run if the batter touches all the bases and scores." It is possible to do so by hitting a fair ball into the outfield and then running very fast: such "inside-the-park home runs" were a staple of baseball before 1920, when balls weren't hit as far and major league baseball stadiums had larger playing areas. Even in 21st century baseball, about a dozen inside-the-park home runs are scored every season, with the most memorable recent one probably hit by Alcides Escobar of the Royals off the very first pitch of the 2015 World Series. This type of play is also a home run (provided the batter is not aided by a defensive error) and is probably more exciting than an out-of-the-park home run.

That aside, the rule that a fair ball hit out of the playing area entitles the batter to a home run has only the loosest connection to cricket boundaries because, within the family of bat-and-ball sports, modern American baseball and cricket are very distant relatives. By 1800 they were recognized as different. For example, in Northanger Abbey by, of course, Jane Austen (published 1817 but written about 20 years earlier), we read the young Catherine Morland "prefer[red] cricket, base-ball, riding on horseback, and running about the country at the age of fourteen, to books." Two different sports, you see.

In fact, the first instance of the term "base ball" is in a 1744 book by John Newbery (published in England and later reprinted in America) entitled A Little Pretty Pocket Book. There is an illustration of men or boys standing by the posts that served as bases in the game under which appears this poem:

Base-Ball

The Ball once struck off,

Away flies the Boy,

To the next destin'd Post,

And then Home with Joy.

Cricket's unlimited back-and-forth running is actually quite distinctive. Older versions of baseball and similar sports (e.g., rounders) do not seem to have featured it ever. Rather, baseball seems always to have required runners to complete a circuit. The goal of the runner is to reach home, as Carlin said, not to run back and forth. And when he reaches home, his turn running the bases ends.

So the fear that without Rule 5.05 the runners would simply circle the bases until the end of time simply does not arise. It is inherent in the logic of baseball, apparently back to the 18th century, that a player can only round the bases and score once before taking a seat and waiting for another opportunity to face the pitcher.

Given that the point of baseball is to round the bases, we may wonder why the reward for hitting a ball out of play is a home run rather than some lesser award. In fact, the famous Knickerbocker Base Ball Club Rules of 1845 specified "one base allowed when a ball bounds out of the field when struck." Since the Knickerbocker rules also defined fair and foul territory, this seems to be a direct ancestor of the "fair ball hit over the fence" rule It's unclear how often this might have occurred. The point has been made by some observers including MLB's official historian John Thorn--and I am inclined to agree--that this rule was made to ensure that the focus of the game was on the contest of fielders and runners. If somehow an unfieldable fair ball were hit, only one base would be allowed.

Baseball historians tend to think that the practice of awarding more bases or a home run came from the evolution of "ground rules" to suit particular playing fields and the award of a home run for a ball hit past the fence and into the outfield stands was practically universal by the 1870s. At that time we start to see rules specifying that a ball hit over a fence at some short distance is just a double, with the implication that a ground rule would exist allowing a home run for a more impressive hit. But keep in mind that until 1929 (American League) and 1931 (National League), the rules did not distinguish between a hit ball that flew over the fence and one that bounced over. The rules were changed to award only a double for the bounce, which is commonly though incorrectly called a "ground rules double." And you still just get one base if a defender hurls the ball out of play.

Eventually "hitting a fair ball over the fence" would come to dominate the game, along with the "third strikes and you're out" rule, but that's another set of evolutions. But in the rule making a fair ball hit out of play a home run probably aren't seeing an attempt to avoid a result found in cricket because the games have just been very different for something like 250 years.

Great sources on the prehistory of baseball are two books by David Block, Baseball Before We Knew It (2005) and Pastime Lost (2019).